Lattice QCD for Hyperon Spectroscopy
David Richards Jefferson Lab
KLF Collaboration Meeting, 12th Feb 2020 Outline
• Lattice QCD - the basics….. • Baryon spectroscopy – What’s been done…. – Why the hyperons? • What are the challenges…. • What are we doing to overcome them… Lattice QCD
• Continuum Euclidean space time replaced by four-dimensional lattice, or grid, of “spacing” a • Gauge fields are represented at SU(3) matrices on the links of the lattice - work with the elements rather than algebra
iaT aAa (n) Uµ(n)=e µ Wilson, 74
Quarks ψ, ψ are Grassmann Variables, associated with the sites of the lattice Gattringer and Lang, Lattice Methods for Work in a finite 4D space-time Quantum Chromodynamics, Springer volume – Volume V sufficiently big to DeGrand and DeTar, Quantum contain, e.g. proton Chromodynamics on the Lattice, WSPC – Spacing a sufficiently fine to resolve its structure Lattice QCD - Summary Lattice QCD is QCD formulated on a Euclidean 4D spacetime lattice. It is systematically improvable. For precision calculations:: – Extrapolation in lattice spacing (cut-off) a → 0: a ≤ 0.1 fm
– Extrapolation in the Spatial Volume V →∞: mπ L ≥ 4
– Sufficiently large temporal size T: mπ T ≥ 10
– Quark masses at physical value mπ → 140 MeV: mπ ≥ 140 MeV – Isolate ground-state hadrons Ground-state masses Hadron form factors, structure functions, GPDs Nucleon and precision matrix elements Low-lying Spectrum
ip x ip x C(t)= 0 (~x, t) †(0) 0 C(t)= 0 e · (0)e · n n †(0) 0 h | | i h | | ih | | i
Need physical “ratios” to fit: mu/d, ms Science 2015 Variational Method
Subleading terms ➙ Excited states Construct matrix of correlators with judicious choice of operators N N Z ⇤Z 1 i j EN t Cij(t, 0) = i(~x, t) j†(~y, 0) = e V3 hO O i 2EN X~x,~y XN Delineate contributions using variational method: solve
(N) (N) C(t)v (t, t0)= N (t, t0)C(t0)v (t, t0).
E (t t ) E(t t ) (t, t ) e N 0 (1 + (e 0 )) N 0 ! O
Can pull out excited-state energies - but pion and nucleon only states stable under strong interactions!
Flavor Spin Orbital Edwards et al., Phys.Rev. D84 (2011) i D! m= 1 = D! x i D! y 074508 p2 Introduce circular basis: ⇣ ⌘ D! m=0 = i D! z Chromomagnetic D! = i D! + i D! . m=+1 p2 x y ⇣ ⌘ [D ,D ] F Straighforward to project to definite spin: J = 1/2, 3/2, 5/2 i j ⌘ ij Positive-parity Baryon Spectrum
[2] 3.0 Dl=1,M
2.5 Hybrid: gluons structural
2.0
1.5
1.0
Dudek, Edwards, PRD85, 054016arXiv:1201.2349 Putting it Together
2000
1500
1000
500
0
Subtract ρ Subtract N
Common mechanism in meson and baryon hybrids: chromomagnetic field with Eg ∼ 1.2 - 1.3 GeV Spectrum is at least as rich as quark model - plus hybrid states across flavor channels in P=+
R. Edwards et al., Phys. Rev. D87 (2013) 054506 Evidence for many charmed Baryons
Bazavov et al, PLB 737, 210 (2014)
All charmed mesons/baryons
Charged charmed mesons/baryons
Strange charmed mesons/baryons
HRG with richer spectrum of states than PDG to describe lattice calculations Thermal Conditions at Freeze-out
Bazavov et al, PRL 113, 072001 (2014)
0.30 µS/µB T=161 MeV T=170 MeV T=166 MeV Including additional 0.25 T=158 MeV T=155 MeV T=162 MeV strange states → T=155 MeV T=149 MeV lower freeze out LQCD: T=155(5) MeV T=152.5 MeV LQCD: T=145(2) MeV T=147 MeV temperature 0.20 T=150 MeV PDG-HRG T=145 MeV QM-HRG 39 GeV (STAR prlim.) 17.3 GeV (NA57) µB/T 0.15 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Can we learn about nature of the states? 4 Hints at structure of Λ(1405)? component within the ⇤(1405). The first term provides the full QCD contribution while the second term subtracts half of the weight of the disconnected sea-quark loop associated with photon couplings to the disconnected loop. Similarly, for the d-quark contribution, we obtain a value of 1 (2d 2 d +d ), 2 n 3 n p and under charge symmetry, the two light quark contributions become equal, conn 1 2 ⇤⇤ µˆ ⇤⇤ = 2u u + u . (12) h | ` | i 2 p 3 p n ✓ ◆ To test the KN model prediction of Eq. (12), we draw on the same set of configurations explored in Ref. [10]. where conn the left-hand side of the equation, ⇤⇤ µˆ ⇤⇤ , was calcu- h | ` | i lated. These calculations are based on the 323 64 full-QCD HallFIG. et 3. al,The Phys. light (u orRev.d) and D strange 95, 054510 quark contributions to the magnetic ⇥ form factor of the ⇤(1405) at Q2 0.16 GeV2/c2 from Ref. [10] are ensembles created by the PACS-CS collaboration [1], made ' presented as a function of the light u- and d-quark masses, indicated by the available through the International Lattice Data Grid (ILDG) squared pion mass, m2 . Sector contributions are for single quarks of unit 3 ⇡ [71]. The ensembles provide a lattice volume of (2.9 fm) charge. The lattice calculations are compared to the predictions of the con- with five different masses for the light u and d quarks, and nected KN model developed herein and summarized in Eq. (12). The vertical dashed line indicates the physical pion mass. The strange form factor results constant strange-quark simulation parameters. We simulate 2 are offset a small amount from the light sector in the m⇡-axis for clarity. the valence strange quark with a hopping parameter of s = 0.13665, reproducing the correct kaon mass in the physical limit [72]. We use the squared pion mass as a renormaliza- loop is significant. In the case where such couplings are in- tion group invariant measure of the quark mass. The scale cluded, the prediction of the KN model is significantly larger is set via the Sommer parameter [73] with r =0.492 fm 0 at ⇤⇤ µˆ ⇤⇤ =(2u + u )/2=1.03(2) µ . Thus, it is h | ` | i p n N [1]. The nucleon magnetic form factors are determined on important for the lattice community to continue to work to- these lattices using the methods introduced in Ref. [68] and wards a determination of these disconnected-loop contribu- refined in Ref. [70], providing values of up =1.216(17) µN tions, particularly for resonances where coupled channel dy- and u = 0.366(19) µ at the lightest pion mass. Results n N namics play an important role. are reported for the lowest nontrivial momentum transfer of The light-quark sector contributions to the magnetic form Q2 0.16 GeV2/c2. ' factor of the ⇤(1405) calculated in lattice QCD [10] have been Lattice QCD results from Ref. [10] for the light- and examined in the context of a molecular KN model in which strange-quark magnetic form factors of the ⇤(1405) are plot- the quark-flow connected contributions to the magnetic form ted as a function of pion mass in Fig. 3. As mentioned earlier, factor have been identified. This enables a quantitative anal- the flavor symmetry present at heavy quark masses is broken ysis of the extent to which the light-quark contributions are as the u and d masses approach the physical point, where consistent with a molecular bound-state description. the strange magnetic form factor drops to nearly zero. The Identification and removal of the quark-flow disconnected light quark sector contribution differs significantly from the contributions to the KN model have been made possible via molecular KN model prediction until the lightest quark mass a recently developed graded-symmetry approach [44]. It is is reached. At this point, the direct matrix element calculation, conn interesting to note that the relative contribution of connected ⇤⇤ µˆ ⇤⇤ of Ref. [10], “⇤(1405) light sector” in Fig. 3, h | ` | i to disconnected contributions is in the ratio 1:2for both agrees with the prediction of the “connected KN model” de- flavor-singlet and flavor-octet representations of the ⇤ baryon. veloped here and summarized in Eq. (12). This agreement Using new results for the magnetic form factors of the nu- confirms that the ⇤(1405) observed in lattice QCD at quark cleon at a near-physical quark mass of m⇡ = 156 MeV, the masses resembling those of Nature is dominated by a molec- connected KN model predicts a light-quark sector contribu- ular KN structure. At the lightest pion mass, the light-quark tion to the ⇤(1405) of 0.63(2) µN , which agrees remarkably magnetic form factor of the ⇤(1405) is [10] well with the direct calculation of 0.58(5) µN from Ref. [10]. conn 2 This confirms that the internal structure of the ⇤(1405) is ⇤⇤ µˆ (Q ) ⇤⇤ =0.58(5) µ , (13) h | ` | i N dominated by a KN molecule. at Q2 0.16 GeV2/c2. The connected KN model of The ⇤(1405) observed in lattice QCD has significant over- ' Eq. (12) predicts lap with local three-quark operators and displays a dispersion relation consistent with that of a single baryon. This implies conn 2 that the KN bound state is localised. Furthermore, it is strik- ⇤⇤ µˆ` (Q ) ⇤⇤ =0.63(2) µN . (14) h | | i ing that the nucleon maintains its properties so well when It is important to note that the shift in the prediction due to the bound. omission of photon couplings to the disconnected sea-quark Future work will focus on the isolation of nearby multi- Spectrum is rich - and strange-quark states essential component
Caveat Emptor! - states are resonances, unstable under strong interactions
“Luscher method” - relate energies shifts at finite volume to infinite-volume scattering amplitudes
R.Briceno,J.Dudek,R.Young, Rev.Mod.Phys. 90 (2018), 025001 Reinventing the quantum-mechanical wheel
Thanks to Raul Briceno (in 1+1 dimensions) L
(x) eipx ⇠
Periodicity: Lpn =2 n Reinventing the quantum-mechanical wheel
Two particles: Reinventing the quantum-mechanical wheel
Two particles:
p* x Reinventing the quantum-mechanical wheel
infinite volume Two particles: scattering phase shift
ip⇤ x +i2 (p⇤) (x) e | | p* x Spectrum: Asymptotic ⇠ wavefunction Reinventing the quantum-mechanical wheel
infinite volume Two particles: scattering phase shift
ip⇤ x +i2 (p⇤) (x) e | | p* x Spectrum: Asymptotic ⇠ wavefunction
Periodicity: Lpn⇤ +2 (pn⇤ )=2⇥n
See Colin Morningstar’s seminar…. Resonant Phase Shift We have treated excitations as stable states - resonances under strong interaction Luscher: finite-volume energy levels to infinite-volume scattering phase shift Hadspec collaboration
Decreasing Pion Mass
Inelastic Threshold , ( p )= ⇤,m Y m(ˆp) (p ) ( p ) O⇥⇥ | | S , ⇤ O⇥ O⇥ m X Xpˆ Wilson, Briceno, Dudek, Edwards, Thomas, arXiv:1507.02599 Transition form factor of ρ
Framework for: • Inelastic scattering • Multi-hadron final states • External currents
Briceno, Hansen and Walker- Loud, PRD 91, 034501 (2015) ⇡
� resonance p Briceno et al., Phys. Rev. D 93, 114508 (2016)
Lattice QCD can calculate what cannot be measured experimentally, e.g. Form factors of resonances. Briceno et al., Phys. Rev. D 100, 034511 (2019) Energy-Momentum Tensor
“Understanding the Glue That Binds Us All: The Next QCD Frontier in Nuclear Physics”
• Quark masses contribute only 1% to mass of proton: binding through gluon confinement • Gluon spin and orbital angular momentum to spin of proton largely unknown 1 1 T = ¯ D + G G G2; P T P = P P /M µ⌫ 4 (µ ⌫) µ↵ ⌫↵ 4 µ⌫ h | µ⌫ | i µ ⌫ (g) Trace Anomaly: T = (1 + ) ¯ + G2 µµ m 2g χQCD Collaboration, ETMC
Yang, this meeting
22 What about Baryons - and hyperons?
4 The theoretical elements are in place……Luka Leskovec
Combinatorics of Wick contractions uud uud much more demanding….
Pion-nucleonLuka scattering Leskovec in the Roper channelet al., arXiv:1806.02363 7 uud uud
1: ON , ON , ON 2000 2: ON , ON N( 2)⇡(2) ud ud 1800 3: ON , ON
1600 N( 1)⇡(1) 4: ON
[MeV] 1400 N(0)⇡(0)⇡(0) 5: ON n
E 1200 P + Fig. 1 Examples of Wick contractions related to two-hadron spectrum in the J =1/2 6: ON channel. The green circles represent the nucleon and grey circles the1000 mesons ⇡ and .The black lines connecting the sink (left side) and the source (right side) are the light quark Combinatorics are 800 propagators. limiting factor 1 2 3 4 5 6
Fig. 4 Spectra for various bases considered. The black dashed lines represent the non- interactingSeeN ⇡nextenergy talk levels by while Colin the dashed Morningstar blue line indicates the energy of the N⇡⇡ threshold.
X En resonant to the ground state and being a linear combination of several states. When nucleon and N⇡ interpolators are considered we find results already discussed in the previous section; however when we replace the N⇡ interpolator with X the N one we find the first excited state has moved to a lower energy; one δ X consistent with the N⇡⇡ threshold energy. When all, nucleon, N⇡ and N , interpolators are included we find that the spectrum contains energy levels X consistent with the nucleon mass, the N⇡⇡ threshold and lowest N⇡ non- interacting scattering energy below 1.65 GeV. No additional energy levels are present. However, as we do not yet know what kind of spectrum we expect X no interaction if the Roper was generated dynamically via coupled channel scattering, we X X X X cannot conclude anything about the Roper based on the spectrum alone. For X X X √s lattice QCD to provide any input on this puzzle, a more complicated and in- X volved analysis is needed. repulsive Comparing our spectrum results with model studies confirms our findings. In particular when comparing to Ref. [19], our lattice energy levels below Fig. 2 Schematic representation of the three possible phase1.65 shift GeV scenarios disagree in with elastic a only scat- bare Roper qqq core interpretation, but are tering and their corresponding finite volume spectra as determined by the L¨uscher method. consistent with results when the N ⇤(1440) resonance is generated dynamically The full green line represents the case where there is no interactionfrom coupling between between the two the hadrons,N⇡, N and ⇡ channels. Our findings also agree red dot-dot-dashed line represents a repulsive interaction betweenwith a recent the two model hadrons study and in Ref. the [20], where the Roper arises as a pole in the dashed blue line a resonant interaction. scattering matrix via dynamical coupling to the N⇡ and N⇡⇡ channels. resonant scattering of a baryon and a meson. A schematic6 Conclusions representation of the three di↵erent situations is shown in Fig. 2. We performed a lattice QCD calculation of the J P =1/2+ channel using In the case of no interaction (case i), shown as the(local) full green single hadron line in and Fig. (non-local) 2, we two hadron interpolating fields on a en- find the spectrum to be made up of scattering levels,semble whose of energies gauge fields correspond with m⇡ 156 MeV [21]. We found that the Roper to cannot arise as a resonance in elastic⇡ N⇡ scattering and is likely a conse- quence of dynamical coupling between several channels. We also note, that
2 2 2 2 Eno int = m + p + m⇡ + p⇡, (6) N N q p Hierarchy of Computations
Capability Computing - Capacity Computing - “Desktop” Computing - Gauge Generation Observable Calculation Physical Parameters
e.g. Summit at ORNL e.g. GPU/KNL cluster at e.g. Mac at your desk JLab, BNL, FNAL SG[U] E t P [U] det M[U]e N C(t)= A e n / 1 n = (U n,G[U n]) hOi N O n n=1 X X Several V, a, T, m π MN (a, m⇡,V) e.g.C(t)= N(~x, t)N¯(0) h i ~ 10% Leadership- X~x Class Resources Computationally Dominant Centered at JLab (not me!)
Major effort at JLab - led by Robert Edwards Important element is speeding up the contractions! Gauge Generation Distillation
Measure matrix of correlation functions: C (t) N (~x, t)N¯ (~y, 0) ij ⌘ h i j i M. Peardon et al., PRD80,054506 (2009) X~x,~y Can we evaluate such a matrix efficiently, for reasonable basis of operators? Introduce ˜ ( ~x, t )= L ( ~x, ~y ) ( ~y, t ) where L is 3D Laplacian n i i Write L (1 /n ) = f ( ) ⇠ ⇠ ⇤ where λi and ξi are eigenvalues ⌘ r i ⇥ and eigenvectors of the Laplacian. i X We now truncate the expansion at i = Neigen where Neigen is sufficient to capture the low- energy physics. Insert between each quark field in our correlation function. ij Perambulators i 1 j ⌧↵ (t, 0) = ⇠⇤ (t)M (t, 0)↵ ⇠ Multi-grid solvers C (t)= i,(pqr)(t) j,(¯pq¯r¯)(0) ⌧ pp¯ (t, 0)⌧ qq¯(t, 0)⌧ rr¯(t, 0) + ... ij ↵ ) ↵¯ ¯ ¯ ⇥ ↵↵¯ ¯ ¯ • Meson correlation functions N3 h Severely constrains i • Baryon correlation functions N4 baryon lattice sizes • Stochastic sampling of eigenvectors - stochastic LaPH – Lower pion mass – Finer isotropic lattice Tanjib Khan, preliminary – Different action Spectroscopy Convergence between Structure
Hybrid-like states Summary • Lattice QCD enables the solution of QCD - it is not modeling QCD! • Lattice calculations have already demonstrated that the importance of a hyperon program: – Spectrum is rich – New states needed to describe phase structure of QCD • Theoretical framework for first-principles calculation is in place: – “Luscher” approach and its extension to multi-channel and inelastic processes – External currents - transition form factors • LQCD can calculate what cannot be determined experimentally • Alignment of theoretical advances, exascale computers, and the software to exploit them! • Convergence of hadron structure and spectroscopy efforts.